On Lie algebras of generalized Jacobi matrices

The first such Lie algebra one may have in mind is the one defined as an inductive limit: let $I$ be a countable set and $I_{1}\subsetneq I_{2}\subsetneq\cdots\subsetneq I_{n}\subsetneq\cdots$, $I=\bigcup_{n}I_{n}$ an increasing series where each $I_{n}$ is a finite subset. The Lie algebra $\mathfrak{gl}_{I}(k)$ is defined by the inductive limit of $\mathfrak{gl}_{I_{m}}(k)\hookrightarrow\mathfrak{gl}_{I_{n}}(k)$ for $m<n$, namely, each element of $\mathfrak{gl}_{I}(k)$ is a finitely supported matrix of infinite size indexed over $I$, i.e., $X=(x_{i,j})_{i,j\in I}$ such that $\sharp\{(i,j)\in I^{2}\,|\,x_{i,j}\neq 0\,\}<\infty$. In particular, for $I=\mathbb{Z}$, we may denote by $\mathfrak{gl}_{\infty}(k)$.

Another one we may also encounter is the Lie algebra of generalized Jacobi matrices defined as follows. A generalized Jacobi matrix is a matrix $M=(m_{i,j})$ indexed over $\mathbb{Z}$ such that there exists a positive integer $N_{M}$ satisfying

The set of such matrices has a structure of associative algebra over $k$ denoted by $J(k)$. We shall denote it by $\mathfrak{g}J(k)$ whenever we regard it as Lie algebra. An original Jacobi matrix is a finite size matrix $M=(m_{i,j})$ such that $m_{i,j}=0$ for any $i,j$ with $|i-j|>1$. The Lie algebra $\mathfrak{gl}_{\infty}(k)$ can be naturally viewed as a subalgebra of $\mathfrak{g}J(k)$.

In the course of studying soliton theory, a non-trivial central extension of the Lie algebra $\mathfrak{g}J(k)$ was discovered (see, e.g., [JM] and [DJM]) and it can be described as follows.

Let $J=\sum_{i\geq 0}E_{i,i}\in\mathfrak{g}J(k)$ be a matrix and let $\Phi:J(k)\rightarrow J(k)$ be the $k$-linear map defined by $X\mapsto JXJ$. It can be checked that, for any $X,Y\in J(k)$, the element $[\Phi(X),\Phi(Y)]-\Phi([X,Y])$ is an element of $\mathfrak{gl}_{\infty}(k)$, i.e., only finitely many matrix entires can be non-zero. Hence, one can define the $k$-bilinear map $\Psi:J(k)\times J(k)\rightarrow k$ by

where $\mathrm{tr}:\mathfrak{gl}_{\infty}(k)\rightarrow k\,;\,X=(x_{i,j})\mapsto\sum_{i\in\mathbb{Z}}x_{i,i}$ is the trace of finitely supported matrices. It turned out that this $\Psi$ is a $2$-cocycle, called Japanese cocycle, i.e.,

1. 1.

   $\Psi(Y,X)=-\Psi(X,Y)$,

2. 2.

   $\Psi([X,Y],Z)+\Psi([Y,Z],X)+\Psi([Z,X],Y)=0$,

for any $X,Y,Z\in J(k)$. Let $\widetilde{\mathfrak{g}J}(k):=\mathfrak{g}J(k)\oplus k\cdot 1$ be the Lie algebra whose Lie bracket $[\cdot,\cdot]^{\prime}$ is given by

As the Lie algebra $\mathfrak{g}J(k)$ is perfect, i.e., $[\mathfrak{g}J(k),\mathfrak{g}J(k)]=\mathfrak{g}J(k)$, the Lie algebra $\mathfrak{g}J(k)$ admits the universal central extension. It was B. L. Feigin and B. L. Tsygan [FT] in 1983 who proved that this central extension $\alpha:\widetilde{\mathfrak{g}J}(k)\rightarrow\mathfrak{g}J(k)$ is universal, namely, $\widetilde{\mathfrak{g}J}(k)$ is perfect and for any central extension $\beta:\mathfrak{a}\rightarrow\mathfrak{g}J(k)$, there exists a morphism of Lie algebras $\gamma:\widetilde{\mathfrak{g}J}(k)\rightarrow\mathfrak{a}$ such that the next diagram commutes:

The extended Lie algebra contains several interesting infinite dimensional Lie algebras as subalgebras. We shall introduce some of them.

1. For an integer $n>1$, let $\widetilde{\mathfrak{g}J_{n}}(k)$ be the subalgebra of $\widetilde{\mathfrak{g}J}(k)$ generated by the matrices $M=(m_{i,j})_{i,j\in\mathbb{Z}}$ satisfying $m_{i+n,j+n}=m_{i,j}$ for any $i,j\in\mathbb{Z}$. This subalgebra is isomorphic to the central extension of $\mathfrak{gl}_{n}(k[t,t^{-1}])$, i.e., the affine Lie algebra

whose commutation relation is given by

where $e_{i,j}\in\mathfrak{gl}_{n}$ is the matrix unit whose $(i,j)$-entry is $1$. An isomorphism between $\widetilde{\mathfrak{gl}_{n}}(k)$ and $\widetilde{\mathfrak{g}J_{n}}(k)$ is given by $e_{i,j}\otimes t^{a}\mapsto\sum_{r\in\mathbb{Z}}e_{i+rn,j+(r+a)n}$.

2. Another important example is the one-dimensional central extension of the Lie algebra $k[t,t^{-1}]\left[\dfrac{d}{dt}\right]$ of algebraic differential operators over $k^{\ast}=k\setminus\{0\}$ that is defined as follows. Set $D=t\frac{d}{dt}$. For any polynomial $f,g\in k[D]$, it can be verified that

Let $\Psi$ be the $2$-cocyle on $k[t,t^{-1}]\left[\dfrac{d}{dt}\right]$ defined by

The Lie algebra $\mathcal{W}_{1+\infty}$ is, by definition, the central extension of $k[t,t^{-1}]\left[\dfrac{d}{dt}\right]$ by the $2$-cocycle $\Psi$, i.e., it is the $k$-vector space

equipped with the Lie bracket given by

It can be shown that there exists morphism of Lie algebras $\mathcal{W}_{1+\infty}\hookrightarrow\widetilde{\mathfrak{g}J}(k)$ satisfying $t^{a}D^{r}\,\mapsto\sum_{i\in\mathbb{Z}}i^{a}E_{i+r,i}$. We remark that the Lie subalgebra $k[t,t^{-1}]D\oplus kC$ of $\mathcal{W}_{1+\infty}$ is isomorphic to the Virasoro algebra.

The first example shows that the Lie algebra $\widetilde{\mathfrak{g}J}(k)$ contains, at least, affine Lie algebras of classical type, i.e., $A_{l}^{(1)}(l\geq 1),B_{l}^{(1)}(l\geq 3),C_{l}^{(1)}(l\geq 2),D_{l}^{(1)}(l\geq 4),A_{2l}^{(2)}(l\geq 1),A_{2l-1}^{(2)},(l\geq 3),D_{l+1}^{(2)}(l\geq 2)$ and $D_{4}^{(3)}$. In addition, the second example shows that Lie algebra $\widetilde{\mathfrak{g}J}(k)$ contains also the Lie algebra $\mathcal{W}_{1+\infty}$ that plays an important role in the KP-hierarchy.

We briefly recall the definitions of Lie algebra (co)homology, Hochschild homology, cyclic homology and (skew-)dihedral homology.

Let $\mathfrak{g}$ be a Lie algebra over a field $k$ of characteristic $0$. From now on, we shall abbreviate the coefficient $k$. The Lie algebra homology $H_{\bullet}(\mathfrak{g})$ is, by definition, the homology of the complex $(\bigwedge^{\bullet}\mathfrak{g},d)$, called the Eilenberg-Chevalley complex, where $\bigwedge^{\bullet}\mathfrak{g}$ is the exterior algebra of $\mathfrak{g}$ and the differential $d$ is given by

The Lie algebra cohomology $H^{\bullet}(\mathfrak{g})$ is by definition, the cohomology of the ‘dual of the complex $(\bigwedge^{\bullet}\mathfrak{g},d)$’ , i.e., $(\mathrm{Hom}_{k}(\bigwedge^{\bullet}\mathfrak{g},k),-{}^{t}d)$. For some basic properties of this homology, see, e.g. [HS]. The homology $H_{\bullet}(\mathfrak{g})$ has a commutative and cocommutative DG-Hopf algebra structure. Its coalgebra structure with counit is induced from the comultiplication

where $\delta:\mathfrak{g}\rightarrow\mathfrak{g}\oplus\mathfrak{g};\,x\,\mapsto(x,x)$ is the diagonal map, and $\sigma$ is the Künneth isomorphism. Hence, the homology $H_{\bullet}(\mathfrak{g})$ is the graded symmetric algebra over its primitive part (cf. [Q]). Recall that an element $x\in H_{\bullet}(\mathfrak{g})$ is said to be primitive if its cocommutative coalgebra structure with counit is induced from the comultiplication which satisfies $\Delta(x)=x\otimes 1+1\otimes x$. For the Lie algebra $\mathfrak{gl}_{\infty}(R)$ over an associative unital $k$-algebra $R$, the primitive part of the homology $H_{\bullet}(\mathfrak{gl}_{\infty}(R))$ had been known by Loday and Quillen [LQ] and independently by B. L. Tsygan [Ts] as follows:

The cohomology ring $H^{\bullet}(\mathfrak{g})$ is naturally endowed with a commutative and cocommutative DG-Hopf algebra structure. Indeed, by the Poincaré duality $H^{n}(\mathfrak{g})\cong H_{n}(\mathfrak{g})^{\ast}$ (full dual), one has $H^{\bullet}(\mathfrak{g})\cong H_{\bullet}(\mathfrak{g})^{\vee}:=\bigoplus_{n}H_{n}(\mathfrak{g})^{\ast}$ (restricted dual).

Now we recall some definitions in homology theory of associative algebras. For detail, see, e.g., [Lod].

Let $R$ be an associative unital $k$-algebra and $M$ be an $R$-bimodule. For $n\in\mathbb{Z}_{\geq 0}$, set $C_{n}(R,M)=M\otimes R^{\otimes n}$. The Hochschild homology $H_{\bullet}(R,M)$ is, by definition, the homology of the complex $(C_{\bullet}(R,M),b)$, where the the differential $b$ is defined by

In particular, for $M=R$, for each $n\in\mathbb{Z}_{>0}$, there is an action of the cyclic group $\mathbb{Z}/(n+1)\mathbb{Z}$ given by

where $x$ is a generator of the group $\mathbb{Z}/(n+1)\mathbb{Z}$. The differential $b$ of the complex $(C_{\bullet}(R,R),b)$ induces a differential on the complex $C_{n}^{\lambda}(R):=R^{\otimes(n+1)}/(1-x)$. The homology of this complex, called Connes’ complex is the so-called cyclic homology $HC_{\bullet}(R)$ of $R$. For some cases, this cyclic homology can be computed with the aid of Connes’ periodicity exact sequence:

Now, assume that $R$ is equipped with a $k$-linear anti-involution $\bar{\cdot}:R\rightarrow R$. One can extend the action of the group $\mathbb{Z}/(n+1)\mathbb{Z}$ on $R^{\otimes(n+1)}$ to the dihedral group $D_{n+1}=\langle x,y|x^{n+1}=y^{2}=1,yxy=x^{-1}\rangle$ by

Let $\mathbf{D}_{n}(R)$ denote the space of coinvariants $(R^{\otimes n+1})_{D_{n+1}}$. If we modify the action of $y$ by $-y$, the resulting coinvariants will be denoted by ${}_{-1}\mathbf{D}_{n}(R)$. The differential $b$ on $C_{\bullet}(R,R)$ induces the differentials on $(R^{\otimes n+1})_{D_{n+1}}$ and ${}_{-1}\mathbf{D}_{n}(R)$, denoted by $\bar{b}$. Their homologies are called dihedral (resp. skew-dihedral) homology of $R$:

For more informations, see, e.g., [Lod].

Let $n>1$ be an integer. We fix a section of $\mathbb{Z}\twoheadrightarrow\mathbb{Z}/n\mathbb{Z}$ and denote its image by $I$. For any matrix $M=(m_{i,j})_{i,j\in\mathbb{Z}}$ and any $i,j\in I$, we set $M_{i,j}:=(m_{i+nk,j+nl})_{k,l\in\mathbb{Z}}$. Then the $k$-linear map

is an isomorphism of Lie algebras $\mathfrak{g}J(k)\longrightarrow\mathfrak{gl}_{n}(J(k))$, and it induces an isomorphism of homologies $H_{\bullet}(\mathfrak{g}J(k))\cong H_{\bullet}(\mathfrak{gl}_{n}(J(k)))$. Taking an inductive limit, we obtain

Hence, the homology $H_{\bullet}(\mathfrak{g}J(k))$ is the commutative and cocommutative DG-Hopf algebra whose primitive part is $HC_{\bullet-1}(J(k))$ by Theorem 3.1. Thus, if we can compute the Hochschild homology $HH_{\bullet}(J(k))$ of the associative algebra $J(k)$, Connes’ periodicity exact sequence (1) allows us to determine the cyclic homology $HC_{\bullet}(J(k))$.

In the rest of this section, we will explain this briefly.

Let $A$ be an associative unital $k$-algebra and let $G$ be a discrete subgroup of the group of $k$-automorphisms of $A$. One can twist the natural product structure on $A\otimes_{k}k[G]$ by $(a\otimes[g])\cdot(b\otimes[h]):=ag(b)\otimes[gh]$, where $a,b\in A$ and $g,h\in G$. The tensor product $A\otimes_{k}k[G]$ equipped with such a twisted product is called twisted group algebra and will be denoted by $A\{G\}$.

Let $M$ be an $A\{G\}$-bimodule.

In our case, $G=\mathbb{Z}$ and $k[\mathbb{Z}]$ is a principal ideal domain, hence its global dimension is at most $1$ which implies that this spectral sequence collapses at $E^{2}$.

By a standard argument, one can show that

Moreover, by direct computation, it can be verified that

This implies $HH_{p}(J(k))\cong HH_{p-1}(k).$

By definition, it follows that

Thus, we obtain

By Connes’ Periodicity long exact sequence (1), one has

Theorem 3.1 and Corollary 3.8 imply

In particular, each graded subspace of $H_{\bullet}(\mathfrak{g}J(k))$ is of finite dimension. Thus, by the Poincaré duality $H^{p}(\mathfrak{g}J(k))\cong H_{p}(\mathfrak{g}J(k))^{\ast}$, one obtains Theorem 1. a) in [FT], i.e., the cohomology ring $H^{\bullet}(\mathfrak{g}J(k))$ has the next description:

Let $R$ be an associative unital $k$-algebra equipped with an $k$-linear anti-involution $\bar{\cdot}$, i.e., $\bar{\bar{s}}=s$ and $\overline{st}=\bar{t}\bar{s}$ for $s,t\in R$. We extend the transpose, also denoted by ${}^{t}(\cdot)$, to $J(R)$ by ${}^{t}(E_{i,j}(r))=E_{j,i}(\bar{r})$, where $E_{i,j}(r)=E_{i,j}r\in J(R)$.

For $l\in\mathbb{Z}$, let $\tau_{l},\tau_{l}^{s}$ be the $k$-linear anti-involutions of the Lie algebra $\mathfrak{g}J(k)$ defined by

We set

Set

We define the anti-involutions $\tau_{B},\tau_{C}$ and $\tau_{D}$ of $\mathfrak{gl}_{\infty}(R)$ as follows:

The Lie algebras

are clearly the stable limit of the Lie algebras $\{\mathfrak{o}_{2l+1}(R)\}_{l\in\mathbb{Z}_{>0}},\{\mathfrak{sp}_{2l}(R)\},\{\mathfrak{o}_{2l}(R)\}_{l\in\mathbb{Z}_{>1}}$, respectively. In a way similar to the case $\mathfrak{g}J(k)$ (cf. Lemma 3.2), one can show

Thanks to Theorem 5.5 in [LP], due to J. L. Loday and C. Procesi, we obtain

It is slightly technical, but one can show (cf. [FI2]) that

Notice that $HD_{p}(k)=k$ for $p\equiv 0\,\text{mod}\,4$ and $HD_{p}(k)=0$ otherwise. Thus, we have